Finitary Group Cohomology and Group Actions on Spheres

TL;DR

This paper proves that certain infinitely generated groups with specific cohomology properties can act freely and orthogonally on spheres, linking algebraic cohomology conditions to geometric group actions.

Contribution

It establishes a new connection between finitary cohomology conditions of groups and their ability to act freely on spheres, extending understanding of group actions.

Findings

Groups with almost everywhere finitary cohomology act freely on spheres.

Infinite locally polycyclic-by-finite groups can have free orthogonal sphere actions.

New link between cohomology properties and geometric actions of groups.

Abstract

We show that if G is an infinitely generated locally (polycyclic-by-finite) group with cohomology almost everywhere finitary, then every finite subgroup of G acts freely and orthogonally on some sphere.